Science:Math Exam Resources/Courses/MATH103/April 2014/Question 09 (b)
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Question 09 (b) 

Consider the iterated map where for . Sketch the cobweb corresponding to the sequence with . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Start by identifying the point on the graph. 
Hint 2 

Next, use the line to find the point . 
Hint 3 

Next, use this point to find . Then repeat this procedure. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The purpose of a cobweb plot is to determine graphically the sequence of points determined by iterating (for some initial condition ). Starting at , our first goal is to determine . To do so, begin by tracing a vertical line from to . Once we have found (see the first hint), we want to determine where lies on the axis (so that we can repeat this procedure). To do so, we begin by finding . Since the  and coordinates of this point agree with each other, it lies on the line . Thus, we sketch the line and determine the point on this line with height . To find the point on the line of height , take the point found in Hint 1 and draw a horizontal line towards the line . The intersection of these two lines is . Once we have found (see Hint 3), we can get determine the point on the axis simply by drawing a vertical line from this point to the axis. We can now repeat this procedure (replacing by , by , etc.). 